Measuring local type might not rule out single field inflation
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based on arXiv : 1104.0244. Measuring local-type might not rule out single-field inflation. Jonathan Ganc Physics Dept., Univ. of Texas July 6, 2011 PASCOS 2011, Univ. of Cambridge. Overview of presentation.

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Measuring local type might not rule out single field inflation

based on arXiv: 1104.0244

Measuring local-type might not rule out single-field inflation.

Jonathan Ganc

Physics Dept., Univ. of Texas

July 6, 2011

PASCOS 2011, Univ. of Cambridge


Overview of presentation

Overview of presentation

  • I will demonstrate that single-field, slow-roll, canonical kinetic-term inflation with a non Bunch-Davies (BD) initial state has an enhanced local-limit bispectrum

  • I will discuss the observed from this scenario in the CMB and whether it could affect the interpretation of a Planck detection of .

Local non-Gaussianity, Jonathan Ganc


Conventional wisdom

Conventional wisdom

  • Single field inflation produces ≈ (5/12) (1 – ns)≈0.01, regardless of potential, kinetic term, or initial vacuum stateCreminelli & Zaldarriaga, 2004

Local non-Gaussianity, Jonathan Ganc


What is

What is… ?

  • … the bispectrum?

the Fourier transform of the three-point function of the curvature perturbation ζ

ζ(x1)

ζ(x2)

ζ(x3)

  • … the squeezed or local limit?

when one of the wavenumbers is much smaller than the other two, e.g. k3≪k1≈k2.

k1

k3

k2

  • … local-type or ?

the best-fit parameter to a target bispectrum by the family of local bispectra, i.e. those generated from a Gaussian field by .


Consider slow roll inflation

Consider slow-roll inflation

  • Canonical action:

  • Assume slow-roll

    ,

  • Write action in terms of (Maldacena 2003):

Local non-Gaussianity, Jonathan Ganc


To quantize

To quantize ζ…

  • … we promote it to an operator :The mode functions , are independent solutions of the classical equation of motion for ζ.

  • The vacuum, slow-roll mode function is

  • We can represent a non-vacuum state by performing a Bogoliubov transformation:new state has occupation number .

vacuum or Bunch-Davies state

⇒,

Local non-Gaussianity, Jonathan Ganc


To calculate the bispectrum

To calculate the bispectrum…

  • … we use the in-in formalism:

  • We find

Local non-Gaussianity, Jonathan Ganc


Bunch davies vs non bunch davies

Bunch-Davies vs. non-Bunch-Davies

Bunch-Davies

(, )

non-Bunch-Davies

in the squeezed limit (

enhancement of in squeezed limit vs. BD!

This effect noticed only recently (Agullo & Parker 2011).

Why was it missed earlier? People expected signal only in folded limit.


What is the observable signal in the cmb from this enhancement

What is the observable signal in the CMB from this enhancement?

  • is calculated from the CMB by fitting the observed angular bispectrum (using transfer functions and projecting onto a sphere) to that predicted by the local bispectrum

  • The angular bispectrum must be calculated numerically.

Local non-Gaussianity, Jonathan Ganc


What we find

What we find

  • If we suppose that across the wavenumbers visible today, Thus, , even for very large .

  • Such a signal is not distinguishable in the CMB.

  • However, it’s larger than predicted by the consistency relation (c.f. ).

Local non-Gaussianity, Jonathan Ganc


Measuring local type might not rule out single field inflation

But…

  • … we’ve glossed over , the phase angle between the Bogoliubov parameters and .

  • Why is this usually OK?

    • Expect to set , at early time by matching mode functions to some non-slow-roll equations.

    • Relative phase will be dominated by exponential factors in mode functions .

    • Thus, we expect . is very large, so oscillates quickly and averages out.

Local non-Gaussianity, Jonathan Ganc


What happens if

What happens if ?

  • Depending on choice of , we can get large positive or negative :

Can achieve for

Local non-Gaussianity, Jonathan Ganc


What about the consistency relation

What about the consistency relation?

  • The consistency relation predicts for single-field models.

  • Here we can have .Is this a counterexample?

Local non-Gaussianity, Jonathan Ganc


Is this a counterexample to the consistency relation

Is this a counterexample to the consistency relation?

  • Initial conditions are set at some time , when is inside the horizon, i.e. .

  • Non-BD terms are multiplied by. On average, the rapidly oscillating term so the above expression ; thus, we get contributions from non-BD terms.

  • In exact local limit, , and we get zero contribution from non-BD terms.

    Consistency relation does hold in exact local limit


The takeaway

The takeaway

  • Slow-roll single-field inflation with an excited initial state can produce an larger than expected.

  • In the more probable case, and still not detectable in the CMB.

  • If we allow the phase angle to be constant, we can get large, detectable .

    The consistency relation is a useful guideline but it holds precisely only in the exact squeezed limit.

    (A similar conclusion is reached in Ganc & Komatsu 2010).

Local non-Gaussianity, Jonathan Ganc


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